now we have an integral \(\int x\cos x\ln\lvert \sin x\rvert dx\). though this may look more complex, it is actually more simple. we can apply integration by parts again. The requirement for integration by parts is that one variable must be easy to differentiate, in this case it is \(x\), and one variable must be easy to integrate, in this case \(\cos x\ln\lvert\sin x\rvert dx\). We know this is easy to integrate as we can see there is a \(\cos x\) and a \(\sin x\), so we can use a simple u-substitution. To make things simple, i shall removed the modulus sigh, but u can factor it back in and u will get 2 answers for the integral